14.3 Interleaved, Recursive & Wrong-Term Series

Interleaved, Recursive & Wrong-Term Series

3. Multiplicative and Mixed Exponential-Additive Series

Many RRB series follow a recursive linear recurrence relation:

$$T_{n+1} = T_n \times k + m$$

Common patterns include:

• $T_{n+1} = T_n \times 2 + 1$
• $T_{n+1} = T_n \times 3 - 2$
• $T_{n+1} = T_n \times n + n$ (variable coefficient and constant)

4. Alternating / Interleaved Series

An alternating series consists of two independent sequences interleaved at alternating positions:

• Odd positions ($T_1, T_3, T_5, T_7, \dots$): Series A (e.g., $+4, +4, +4$)
• Even positions ($T_2, T_4, T_6, T_8, \dots$): Series B (e.g., $\times 2, \times 2, \times 2$)

Interleave before inventing a complicated formula

An alternating series can contain two simpler sequences. For 12, 45, 15, 40, 18, 35, separate the odd positions 12, 15, 18 from the even positions 45, 40, 35. The odd stream increases by 3 and the even stream decreases by 5. The next term is 21, because the seventh position belongs to the odd stream. The following term is 30. Extending the wrong stream is a common indexing error.

A mixed recurrence applies operations in a stated order. If a rule is multiply by two and add successive odd numbers starting at one, the sequence beginning at 5 is 5, 11, 25, 55, 117, 243. Check each step: 5 times 2 plus 1 gives 11; 11 times 2 plus 3 gives 25; then add 5, 7 and 9 after the respective doublings. Adding before multiplying produces a different sequence.

A clean wrong-term example

Suppose the same intended rule is stipulated, but the printed list is 4, 11, 25, 55, 117, 243. Working backward, the second term requires the first to satisfy twice the first plus one equals 11. The first term should therefore be 5. Replacing 4 with 5 fixes the entire list. Changing a later term merely to repair one local transition would leave other transitions inconsistent.

An internal incorrect term affects the difference on each side. For the intended list 2, 5, 10, 17, 26, replacing 10 by 12 changes the observed first differences from 3, 5, 7, 9 to 3, 7, 5, 9. The two central differences are disturbed in opposite directions. At an endpoint, only one first difference is affected, so a rule saying every error corrupts exactly two differences needs that qualification.

Keep complexity under control

Try constant differences, ratios, familiar powers, alternating streams and short recurrences before high-degree polynomial fitting. These are alternative hypotheses to test, not a compulsory hierarchy imposed by RRB. Do not spend most of a timed test defending an elaborate rule with exceptions. If the intended structure is unclear, mark the item and return after easier questions.

When reviewing, write the corrected sequence and every transition in one line. Check the predicted next term against the same rule, rather than switching methods at the final step. For an alternating pattern, label positions explicitly; for a recurrence, label which additive or multiplicative parameter belongs to each transition. This makes errors in indexing visible.

Verify a correction independently

Consider the intended sequence with rule twice the previous term minus one: 4, 7, 13, 25, 49. If the printed sequence is 4, 7, 15, 25, 49, test each candidate correction against both neighboring transitions. The third term must be 13 because 2 × 7 - 1 = 13 and 2 × 13 - 1 = 25. Replacing 25 instead would still leave the transition from 7 to 15 unexplained. The repair must remove every violation of the stated rule.

For an interleaved exercise, write position labels before combining the streams. In 2, 30, 4, 27, 8, 24, 16, ?, the odd positions double while the even positions decrease by three. The eighth term is 21. A prediction of 32 extends the odd stream into an even position and answers a different question.

These examples explicitly state or exhibit the intended simple model. A sufficiently elaborate polynomial could fit a different finite list, so uniqueness in an aptitude exercise is a matter of the supplied constraints and consistent intended pattern. During review, record both the generating rule and the index of the altered or missing term.

Separate the correction target from the next-term target

Consider 1, 4, 9, 18, 25, 36, with the intended consecutive-square rule. The fourth term is wrong and should be 16. If asked for the next term after correcting the list, the answer is 49. Selecting 16 for a next-term question identifies the error but does not answer the requested task.

After replacing 18 by 16, the first differences are 3, 5, 7, 9 and 11. They increase by two throughout. Before correction, the differences around the fourth term are 9 and 7, showing the local disturbance. The neighboring unaffected differences help locate the error, while the stated square model confirms its corrected value. Keep the reported wrong term, replacement value and next term in separate columns during revision.